The three general types of boiler you're likely to hear mentioned by heating engineers are:
Condensing boilers are a relatively new type of boiler they are a lot more eficiant at extracting the heat energy in the gas than non-condensing boilers. This means they burn less gas for the same amount of heating, leading to slightly lower fuel bills and slightly less carbon dioxide emitted by the boiler into the atmosphere. Carbon dioxide is generally acknowledged to be a 'greenhouse gas', and is widely believed to contribute to 'global warming'. Condensing boilers have now been made compulsory and must be fitted by a gas installer to comply with building regulations.
Combi boilers do two main jobs. The first is to supply heating to your radiators and the second is to supply hot water on demand to a hot tap when it is turned on. The cold water passes through a plate heat exchanger which is heating by the boiler to give you hot water from your taps. A non-combi boilers heats water which in turn heat a copper cylinder which stores the hot water for later use.
System boilers are boilers designed to be installed along with a hot water cylinder. They can be installed with water tanks in the loft or as a pressurised system. In some cases they have an expansion vessel and the circulating pump built into them. This can save time and money on the installation but are around the same size and price as an equivalent combi boiler.